Nonlinear Equations Riccati Equation

Very few nonlinear equations yield analytical solutions, so graphical or trial-error solution methods are often used. There are a few nonlinear finite difference equations, which can be reduced to linear form by elementary variable transformation. Foremost among these is the famous Riccati equation 

The remainder can be made linear by dividing by z +iZ and introducing a new variable 

This is an elementary first order linear equation with forcing by a constant. The characteristic root is simply 

The cascade shown in Fig. 5.1 could also represent a plate-to-plate distillation operation if we denote y as solute mole fraction in vapor and x is the fraction in liquid. For constant molar flow rates, L and V are then constant. Now, j represents hot vapor feed, and Xq represents desired product recycle, which is rich in the volatile solute. For the high concentration expected, the relative volatility (a) is taken to be constant, so the equilibrium relation can be taken as 

This is obtained from the usual definition of relative volatility for binary systems 

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Nonlinear Difference Equations Riccati Difference Equation The Riccati equation yx-t-iyx + 1 + byx + c = 0 is a nonhnear... 

A nonlinear equation, which arises in both continuous and staged (i.e., finite difference) processes, is Riccati s equation... 

It is straightforward to see that the superpotential nonlinear first-order differential equation, a Riccati equation. 

Consider the reflection of a normally incident time-harmonic electromagnetic wave from an inhomogeneous layered medium of unknown refractive index n(x). The complex reflection coefficient r(k,x) satisfies the Riccati nonlinear differential equation [2] ... 

In this chapter, we develop analytical solution methods, which have very close analogs with methods used for linear ODEs. A few nonlinear difference equations can be reduced to linear form (the Riccati analog) and the analogous Euler-Equidimensional finite-difference equation also exists. For linear equations, we again exploit the property of superposition. Thus, our general solutions will be composed of a linear combination of complementary and particular solutions. 

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